The use of numerical analysis to determine attenuation of radiation traversing or scattering out of an object has been used since the 1950's. This science has always had its limits and its use remains to the present a considerable challenge. Since powerful computers are now readily available at reasonable price, however, and that the associated mathematics are better understood, numerical calculation methods have been more commonly used to generate density images of objects.
Examples of x-ray imaging systems and methods belonging to the prior art and using numerical analyses are described in the following documents:    U.S. Pat. No. 4,785,401 issued to G. Harding et al. on Nov. 15, 1988;    U.S. Pat. No. 4,887,285 issued to G. Harding et al. on Dec. 12, 1989;    U.S. Pat. No. 5,070,455 issued to J. R. Singer et al. on Dec. 3, 1991;    U.S. Pat. No. 5,696,806 issued to L. Grodzins et al. on Dec. 9, 1997;    U.S. Pat. No. 5,729,582 issued to Y. S. Ham et al. on Mar. 17, 1998;    U.S. Pat. No. 5,838,758 issued to K. D. Krug et al. on Nov. 17, 1998;    U.S. Pat. No. 5,930,326 issued to P. Rothschild et al. on Jul. 27, 1999;    U.S. Pat. No. 6,018,562 issued to P. D. Willson on Jan. 25, 2000;    U.S. Pat. No. 6,052,433 issued to Yong-Sheng Chao on Apr. 18, 2000;    U.S. Pat. No. 6,556,653 issued to E. Hussein on Apr. 29, 2003;
In addition to the above-mentioned prior art documents, the following US Patents are particularly relevant herein as they describe methods and installations using Compton scattering, elaborate mathematical calculations and combination of one radiation source with two detectors or two radiation sources with two detectors. These documents are:    U.S. Pat. No. 3,202,822 issued to P. Kehler on Aug. 24, 1965;    U.S. Pat. No. 3,809,904 issued to R. L. Clarke et al. on May 7, 1974;    U.S. Pat. No. 3,840,746 issued to P. Kehler on Oct. 8, 1974;    U.S. Pat. No. 4,558,220 issued to H. B. Evans on Dec. 10, 1985;    U.S. Pat. No. 4,768,214 issued to P. J. Bjorkholm on Aug. 30, 1988;    U.S. Pat. No. 4,956,856 issued to G. Harding on Sep. 11, 1990;    U.S. Pat. No. 5,729,582 issued to Y. S. Ham et al. on Mar. 17, 1998;    U.S. Pat. No. 5,970,116 issued to S. Dueholm et al. on Oct. 19, 1999;    U.S. Pat. No. 6,094,470 issued to S. Teller on Jul. 25, 2000;    U.S. Pat. No. 6,563,906 issued to E. Hussein et al. on May 13, 2003;
Although several numerical analysis methods are available in the prior art, the forward-inverse numerical analysis algorithm is believed to be the most practical one for radiation-scattering imaging of thick objects. However, it is also believed that this forward-inverse numerical analysis algorithm has been generally overlooked in the past. This numerical analysis method is briefly explained as follows.
Mathematically, radiation-scattering imaging is treated as an inverse problem. To define the inverse problem, one must define the direct or forward problem. The forward problem is the mapping of a set of theoretical parameters into a set of experimentally measurable results. Solving the forward problem is then effected to obtain computed results from given parameters. Obtaining the actual parameters from the detector responses is called solving the inverse problem.
Numerical analysis of radiation-scattering imaging using a forward-inverse numerical analysis algorithm has been previously described by E. M. A. Hussein, D. A. Meneley, and S. Banerjee, in an article entitled: “On the Solution of the Inverse Problem or Radiation Scattering Imaging” published in 1986 in a publication entitled: Nuclear Science and Engineering, issue 92, pages 341–349.
Imaging using scattered radiation, records events that take place deep within an object. In essence, the scattering signal is modulated by the attenuation of radiation as it travels toward the point of scattering and then as it returns to a detector. This renders a nonlinear inverse problem, since the attenuation process is exponential in nature, while the scattering process is linear.
The challenge posed by this non-linearity is best demonstrated by considering the forward problem of scattering from a single voxel. While referring to FIG. 1, it will be appreciated that the scattering of a pencil beam of radiation from a source i, of some incident energy E, is scattered by an angle of 90° within an object having a single voxel j, and is returned to a detector k with an energy E′. The detector response sijk can be expressed as:
                              s          ijk                =                              c            ijk                    ⁢                      exp            ⁡                          [                                                -                                      μ                    ⁡                                          (                      E                      )                                                                      ⁢                                  x                  2                                            ]                                ⁢          ρ          ⁢                                          ⁢                      exp            ⁡                          [                                                -                                      μ                    ⁡                                          (                                              E                        ′                                            )                                                                      ⁢                                  y                  2                                            ]                                                          (        1        )            where ρ refers to density; μ refers to the attenuation coefficient (which is a function of E and ρ); x is the distance travelled by the incident radiation within the voxel along the direction of the incident beam, y is the distance travelled by the scattered beam away from the incident beam, and cijk is a pre-determined system constant that depends of the probability of scattering, source-voxel-detector geometric arrangement, source intensity, detector efficiency, etc.
To further simplify the problem, let x=y, replace μ(E) and μ(E′) with some average value between the two, and assume that μ and ρ are physically related such that μ=σρ, where σ is a known parameter. Equation (1) for a single voxel can now be written as:
                              s          ijk                =                                            c              ijk                        σ                    ⁢          μ          ⁢                                          ⁢                      exp            ⁡                          [                                                -                  μ                                ⁢                                                                  ⁢                x                            ]                                                          (        2        )            This equation represents the “forward” problem that relates the problem parameters, attenuation coefficient, to a measurable detector response.
This forward model demonstrates clearly the competition between the linear term (scattering), which increases with increasing μ and the exponential term (attenuation) which declines in value with increasing μ.
Therefore, Sijk increases with μ until it reaches a maximum value at μ=x−1, then it decreases with increasing μ. Upon solving the inverse problem to find the value for μ at a given value for sijk two solutions are possible. This simple equation is plotted on a graph in FIG. 2 for various values of x. A dotted line 100 extending across the graph shows the two values of μ for each value of sijk.
The first solution is for μ<x−1 and the other solution at μ>x−1. A single solution only exists at μx=1, where sijk reaches it maximum value, at which point
      x    =          1      μ        ,i.e. equal to one mean-free-path (mfp).
It will be appreciated that the linear component of Equation (2) is dominant when the distance travelled is less than the average distance travelled by radiation (<1 mfp), while the exponential component is dominant when the distance travelled is greater than the mfp.
Due to the non-linearity of the forward-inverse problem, direct solution is impractical. Therefore, one or more iterations using adjusted computed results may be required to obtain a solution with minimum differences between the measured results and the computed results. However, an iterative solution of Equation (1) can converge to either one of the two possible solutions. When dealing with more than one voxel to reconstruct a realistic image, oscillation between the two possible values of μ at each voxel can result in an unstable iterative process. It will be appreciated that this problem becomes much more complex when a fan beam is utilized, wherein each detector measurement contains attenuation and scattering from several voxels within the field of view of the detector. Therefore, a forward-inverse numerical analysis algorithm applied to fan beams has been considered in the past as been practically unsolvable by any conventional ways.
Because of the diverging nature of radiation, a pencil beam per se is a theoretical expression only. It is believed that a pencil beam as described in the prior art, is in reality, a narrow cone beam and the attenuation and scattering contributed by voxels near the axis of the beam cannot be ignored. Therefore it is believed that a forward-inverse numerical analysis algorithm could have been used in the past in the field of density-imaging of objects, but with many approximations, assumptions and omissions.
As such, it is believed that a need exists for a new method and installation for single-side x-ray density-imaging of objects using a forward-inverse numerical analysis algorithm, wherein non-linearity of the problem is not an obstacle.